The Southampton attacker was one of four Italians to miss from the spot, the game having ended 1-1 after 120 minutes, with Simone Zaza, Leonardo Bonucci and Matteo Darmian also failing to find the net.
Pelle was heavily criticised for his antics ahead of his penalty, making a Panenka-gesture toward Germany goalkeeper Manuel Neuer, and he has now apologised for his miss.
"It is not important to me now that it was a good tournament for Italy overall. I am very sad and I would like to apologise to all of Italy. I want to say sorry to my team-mates, to the coach, to the fans and to anyone else who wished me the best," Pelle told reporters.
"I had just one job and that was to score the penalty. I would have become a phenomenon had I done that. I was a nobody when I arrived at the Euros and I am still a nobody now. But I have always given my all for the jersey of Italy.
"I did not want to disrespect Neuer. I wanted to deceive him. I saw that he dived before the penalty takers had hit their shot and he seemed to be flying to the corners. I tried to make him stay in the middle of his goal a bit longer by making the Panenka-gesture."